I could not reproduce this issue using the "Kitchen Sink >> User Interface >> Forms" example. With Chrome 18.0.1025469 (Android 4.0) or iPad 3 (iOS 6.0.1). On what version of Chrome and iOS are you seeing this issue.

The problem is only occurring with Chrome. (Sorry to have reported iOS as well.)
Chrome Version 25.0.1364.97 m
Windows 7, but also seen on RDP to Window Server 2012.

I click the DatePicker. The picker slides in from the bottom. I use the mouse to move the month to the next month. I click the Done button. Nothing happens in the DatePicker.

However, if after moving the month with the mouse, I then click on the new month inside the band across the middle, then the change is effective. Also works if click on a new month outside the band, that also moves it to the band and the change is effective.